$$f(x) =\dfrac{e^x}{1+e^x}$$
I know we can find points of inflection using the second derivative test. The second derivative for the function above is $$f''(x) = \dfrac{e^x(1-e^x)}{(e^x+1)^3}$$ I have ...

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it:
Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...

I need a calculus book that has all the details or is closest to it. If one book for calculus would not be enough to understand concepts, kindly recommend books that don't have overlapping concepts.
...

For $n$ tending to infinity find the following limit $$2^n/n!$$
i have a feeling that it is multiplication of many numbers with the last one turning to 0 but the 1st one is finite so limit should be 0 ...

Assuming that I have a linear ODE without any singularities over the complex numbers
$$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$
Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part.
$$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...

A problem given in Spivak's Calculus text is to show that a function $f:[a,b]\to \mathbb{R}$ cannot have a strict local maximum at each point. I will sketch the proof below the fold.
My question is: ...

How to find
$$\int\dfrac{dx}{1+x^{2n}}$$
where $n \in \mathbb N$?
Remark
When $n=1$, the antiderivative is $\tan^{-1}x+C$. But already with $n=2$ this is something much more complicated. Is there a ...

Let ${a_{n}}$ be a convergent sequence in $\mathbb Z$. Is it true that $\lim_{n\rightarrow \infty}{a_{n}}\in \mathbb Z$?
Remark
This is false if $\mathbb Z$ is replaced by $\mathbb Q$, because then ...